Let f(x) = left{ {begin{array}{*{20}{c}}
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Question

(d) For Rolle&rsquo;s theorem to be applicable to $f,$ for $x \in [0,\,1]$, we should have

$(i)$ $f(1) = f(0)$,

$(ii)$ $f$ is continuous for $x

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(d) For Rolle&rsquo;s theorem to be applicable to $f,$ for $x \in [0,\,1]$, we should have

$(i)$ $f(1) = f(0)$,

$(ii)$ $f$ is continuous for $x \in [0,\,1]$ and $f$ is differentiable for $x \in (0,\,1)$

From $(i),$ $f(1) = 0$, which is true.

From $(ii),$ $0 = f(0) = f({0_ + }) = \mathop {\lim }\limits_{x 	o {0_ + }} {x^\alpha }\ln x$

Which is true only for positive values of $\alpha $, thus  $(d)$  is correct.

Let $f(x) = \left\{ {\begin{array}{*{20}{c}}
{{x^2}\ln x,\,x > 0} \\
{0,\,\,\,\,\,\,\,\,\,\,\,\,\,x = 0}
\end{array}} \right\}$, Rolle’s theorem is applicable to $ f $ for $x \in [0,1]$, if $\alpha = $

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